Monotone substochastic operators and a new Calderón couple

Karol Leśnik. "Monotone substochastic operators and a new Calderón couple." Studia Mathematica 227.1 (2015): 21-39. <http://eudml.org/doc/285740>.

@article{KarolLeśnik2015,
abstract = {An important result on submajorization, which goes back to Hardy, Littlewood and Pólya, states that b ⪯ a if and only if there is a doubly stochastic matrix A such that b = Aa. We prove that under monotonicity assumptions on the vectors a and b the matrix A may be chosen monotone. This result is then applied to show that $(\widetilde\{L^\{p\}\},L^\{∞\})$ is a Calderón couple for 1 ≤ p < ∞, where $\widetilde\{L^\{p\}\}$ is the Köthe dual of the Cesàro space $Ces_\{p^\{\prime \}\}$ (or equivalently the down space $L^\{p^\{\prime \}\}_\{↓\}$). In particular, $(\widetilde\{L¹\},L^\{∞\})$ is a Calderón couple, which gives a positive answer to a question of Sinnamon [Si06] and complements the result of Mastyło and Sinnamon [MS07] that $(L^\{∞\}_\{↓\},L¹)$ is a Calderón couple.},
author = {Karol Leśnik},
journal = {Studia Mathematica},
keywords = {substochastic operator; doubly stochastic matrix; Calderón couple},
language = {eng},
number = {1},
pages = {21-39},
title = {Monotone substochastic operators and a new Calderón couple},
url = {http://eudml.org/doc/285740},
volume = {227},
year = {2015},
}

TY - JOUR
AU - Karol Leśnik
TI - Monotone substochastic operators and a new Calderón couple
JO - Studia Mathematica
PY - 2015
VL - 227
IS - 1
SP - 21
EP - 39
AB - An important result on submajorization, which goes back to Hardy, Littlewood and Pólya, states that b ⪯ a if and only if there is a doubly stochastic matrix A such that b = Aa. We prove that under monotonicity assumptions on the vectors a and b the matrix A may be chosen monotone. This result is then applied to show that $(\widetilde{L^{p}},L^{∞})$ is a Calderón couple for 1 ≤ p < ∞, where $\widetilde{L^{p}}$ is the Köthe dual of the Cesàro space $Ces_{p^{\prime }}$ (or equivalently the down space $L^{p^{\prime }}_{↓}$). In particular, $(\widetilde{L¹},L^{∞})$ is a Calderón couple, which gives a positive answer to a question of Sinnamon [Si06] and complements the result of Mastyło and Sinnamon [MS07] that $(L^{∞}_{↓},L¹)$ is a Calderón couple.
LA - eng
KW - substochastic operator; doubly stochastic matrix; Calderón couple
UR - http://eudml.org/doc/285740
ER -